Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation

Abstract

Using the weak-noise theory, we evaluate the probability distribution P(H,t) of large deviations of height H of the evolving surface height h(x,t) in the Kardar-Parisi-Zhang equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height H at time t. We argue that the tails of P behave, at arbitrary time t>0, and in a proper moving frame, as -lnP∼|H|^{5/2} and ∼|H|^{3/2}. The 3/2 tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.

Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation: Starting from a parabola

“…At L → ∞ one has Φ = 2 (the horizontal dashed line) in agreement with Refs. [23,25]. Notice the non-analytic w 1/3 behavior of f (w) at w → 0.…”

“…(23) and (24) into Eqs. (16) and (17), we obtain two coupled equations for r(t) and a(t):ṙ = −ra andȧ = −a 2 − (32/9)r 3 [25]. Their first integral can be written as a = ±(8/3)r √ r − r * , where r * ≡ r(t = 1/2).…”

“…These equations describe a non-stationary inviscid flow of a compressible gas with density ρ, velocity V , and negative pressure p(ρ) = −ρ 2 /2 [25,43]. The problem should be solved subject to the condition…”

“…The long-time regime has traditionally attracted great interest [2][3][4][5][6], but the short-time regime is also interesting [21][22][23][24]. Indeed, at short times one observes, for both flat and sharp-wedge initial conditions, crossover of the full one-point height statistics from the Edwards-Wilkinson universality class to the KPZ universality class as one moves away from the body of the distribution P(H) to its strongly asymmetric tails [20,21,[23][24][25].…”

“…This approach was taken in Refs. [21][22][23]25] which studied the short-time asymptotics of P(H, t) when starting the process from a flat interface. In these works the probability distribution P(H, t) was evaluated by using the weaknoise theory (WNT) of Eq.…”